Abstract: Using the tools introduced in [Breckner, B. E., and W. A. F. Ruppert, J. Lie Theory 11 (2001), 559--604], we investigate topological semigroup compactifications of closed connected submonoids with dense interior of Sl(2,R). In particular, we show that the growth of such a compactification is always contained in the minimal ideal, and describe the subspace of all minimal idempotents (typically a two-cell) and the maximal subgroups (these are always isomorphic with a compactification of R). For a large class of such semigroups we give explicit constructions yielding all possible topological semigroup compactifications and determine the structure of the compactification lattice.